8,469 research outputs found
On the editing distance of graphs
An edge-operation on a graph is defined to be either the deletion of an
existing edge or the addition of a nonexisting edge. Given a family of graphs
, the editing distance from to is the smallest
number of edge-operations needed to modify into a graph from .
In this paper, we fix a graph and consider , the set of
all graphs on vertices that have no induced copy of . We provide bounds
for the maximum over all -vertex graphs of the editing distance from
to , using an invariant we call the {\it binary chromatic
number} of the graph . We give asymptotically tight bounds for that distance
when is self-complementary and exact results for several small graphs
Characterization and Efficient Search of Non-Elementary Trapping Sets of LDPC Codes with Applications to Stopping Sets
In this paper, we propose a characterization for non-elementary trapping sets
(NETSs) of low-density parity-check (LDPC) codes. The characterization is based
on viewing a NETS as a hierarchy of embedded graphs starting from an ETS. The
characterization corresponds to an efficient search algorithm that under
certain conditions is exhaustive. As an application of the proposed
characterization/search, we obtain lower and upper bounds on the stopping
distance of LDPC codes.
We examine a large number of regular and irregular LDPC codes, and
demonstrate the efficiency and versatility of our technique in finding lower
and upper bounds on, and in many cases the exact value of, . Finding
, or establishing search-based lower or upper bounds, for many of the
examined codes are out of the reach of any existing algorithm
Note on the upper bound of the rainbow index of a graph
A path in an edge-colored graph , where adjacent edges may be colored the
same, is a rainbow path if every two edges of it receive distinct colors. The
rainbow connection number of a connected graph , denoted by , is the
minimum number of colors that are needed to color the edges of such that
there exists a rainbow path connecting every two vertices of . Similarly, a
tree in is a rainbow~tree if no two edges of it receive the same color. The
minimum number of colors that are needed in an edge-coloring of such that
there is a rainbow tree connecting for each -subset of is
called the -rainbow index of , denoted by , where is an
integer such that . Chakraborty et al. got the following result:
For every , a connected graph with minimum degree at least
has bounded rainbow connection, where the bound depends only on
. Krivelevich and Yuster proved that if has vertices and the
minimum degree then . This bound was later
improved to by Chandran et al. Since , a
natural problem arises: for a general determining the true behavior of
as a function of the minimum degree . In this paper, we
give upper bounds of in terms of the minimum degree in
different ways, namely, via Szemer\'{e}di's Regularity Lemma, connected
-step dominating sets, connected -dominating sets and -dominating
sets of .Comment: 12 pages. arXiv admin note: text overlap with arXiv:0902.1255 by
other author
Distance-regular graphs
This is a survey of distance-regular graphs. We present an introduction to
distance-regular graphs for the reader who is unfamiliar with the subject, and
then give an overview of some developments in the area of distance-regular
graphs since the monograph 'BCN' [Brouwer, A.E., Cohen, A.M., Neumaier, A.,
Distance-Regular Graphs, Springer-Verlag, Berlin, 1989] was written.Comment: 156 page
A Breezing Proof of the KMW Bound
In their seminal paper from 2004, Kuhn, Moscibroda, and Wattenhofer (KMW)
proved a hardness result for several fundamental graph problems in the LOCAL
model: For any (randomized) algorithm, there are input graphs with nodes
and maximum degree on which (expected) communication rounds are
required to obtain polylogarithmic approximations to a minimum vertex cover,
minimum dominating set, or maximum matching. Via reduction, this hardness
extends to symmetry breaking tasks like finding maximal independent sets or
maximal matchings. Today, more than years later, there is still no proof
of this result that is easy on the reader. Setting out to change this, in this
work, we provide a fully self-contained and proof of the KMW
lower bound. The key argument is algorithmic, and it relies on an invariant
that can be readily verified from the generation rules of the lower bound
graphs.Comment: 21 pages, 6 figure
The (a,b,s,t)-diameter of graphs: a particular case of conditional diameter
The conditional diameter of a connected graph is defined as
follows: given a property of a pair of
subgraphs of , the so-called \emph{conditional diameter} or -{\em diameter} measures the maximum distance among subgraphs satisfying
. That is, In this paper we consider the conditional diameter in
which requires that for all , for all , and for some integers and
, where denotes the degree of
a vertex of , denotes the minimum degree and the
maximum degree of . The conditional diameter obtained is called
-\emph{diameter}. We obtain upper bounds on the -diameter by using the -alternating polynomials on the mesh of
eigenvalues of an associated weighted graph. The method provides also bounds
for other parameters such as vertex separators
- …